Auto-Resizing objects in BGE when I press play? How do I fix this
Select the parent and hit CTRL + A and select "Scale" 

EDIT: An explanation: This only happens when using a a vertex parent.

An ordinary parenting causes the child to follow scale translation and rotation of the parent's Origin point. 

A vertex parent in blender game engine copies the same information, but from a vertex, not the objects origin point. For reasons i do not know, if you edit the location of the vertex in edit mode, the child doesn't appear to notice while when you press play, and only appears to update when you apply the location of the parent. That is a best of an explanation i can give, if you know more, then please edit this post and add your knowledge.